.. index:: Bimodal Instability .. index:: Bipolar Model Bipolar Model ========================= Bimodal in this context means :highlight-text:`two frequencies` [Samuel1996]_. With neutrino coherent scattering, the neutrino state consists of two frequencies. An example of such intability happens in a system composed of equal amounts of neutrinos and antineutrinos. Flavour transform occurs due to .. math:: \nu_e + \bar{\nu_e} \leftrightarrow \nu_x + \bar{\nu_x}. Vacuum mixing angle triggers the flavour instability. Neutrino oscillations has a small amplitude inside a SN core (suppressed by matter effects) [Wolfenstein1978]_, which basically pins down the flavour transformation. As the neutrinos reaches a furthur distance, matter effect could drop out. Neutrino self-interaction becomes more important. [Samuel1996]_ considers a system of neutrinos and antineutrinos with only vacuum and neutrino self-interactions. The neutrinos and antineutrino forms a bipolar vector in flavor isospin space. The flavor isospin of neutrinos and that of antineutrinos are coupled. [Duan2013]_ decomposed the system into "normal modes" of the flavor isospin. The bipolar system is discussed in details in this paper. In a two beam model, the length of one of the perturbations can be discribed using an equation .. math:: \ddot {\tilde q_+} \approx - \eta (\eta + \mu) \tilde q_+, where :math:`\eta=\pm 1` deterimines the hierarchy, :math:`\mu=2\sqrt{2}G_F \lvert \omega_0 \rvert^{-1} n_{\mathrm {tot}}`. We find out from the equation that normal hierarchy (NH, :math:`\eta=1`) doesn't have instabilities, but inverted hierarchy (IH, :math:`\eta=-1`) has instabilities with growth rate :math:`\sqrt{\mu-1}`, if :math:`\mu>1`. Linear Stability Analysis -------------------------------- The equation of motion is .. math:: i\partial_t \rho =& \left[ -\frac{\omega_v}{2} \cos2\theta \sigma_3 + \frac{\omega_v}{2}\sin 2\theta \sigma_1 - \mu \alpha \bar \rho , \rho\right] \\ i\partial_t \bar\rho =& \left[ \frac{\omega_v}{2} \cos2\theta \sigma_3 - \frac{\omega_v}{2}\sin 2\theta \sigma_1 + \mu \rho , \bar\rho\right]. For the purpose of linear stability analysis, we assume that .. math:: \rho =& \frac{1}{2}\begin{pmatrix} 1 & \epsilon \\ \epsilon^* & -1 \end{pmatrix} \\ \bar\rho =& \frac{1}{2}\begin{pmatrix} 1 & \bar\epsilon \\ \bar \epsilon^* & -1 \end{pmatrix}. Plug them into equation of motion and set :math:`\theta=0`, we have the linearized ones, .. math:: i\partial_t \begin{pmatrix} \epsilon \\ \bar\epsilon \end{pmatrix} = \frac{1}{2}\begin{pmatrix} -\alpha \mu - \omega_v & \alpha \mu \\ -\mu & \mu + \omega_v \end{pmatrix}\begin{pmatrix} \epsilon \\ \bar\epsilon \end{pmatrix}. To have real eigenvalues, we require .. math:: (-1+\alpha)^2 \mu^2 + 4(1+\alpha)\mu \omega_v + 4 \omega_v^2 < 0, which is reduced to .. math:: \frac{ -2\omega_v (1+\alpha) - 4\sqrt{ \alpha } \lvert \omega_v \rvert }{ (1-\alpha)^2 } < \mu < \frac{ -2\omega_v (1+\alpha) + 4\sqrt{ \alpha } \lvert \omega_v \rvert }{ (1-\alpha)^2 }, which is simplified to .. math:: \sqrt{ -2\omega_v }{ (1-\sqrt{\alpha})^2 } < \mu < \sqrt{ -2\omega_v }{ (1+\sqrt{\alpha})^2 }, assuming normal hierarchy, i.e., :math:`\omega_v > 0`. We immediately notice that this can not happen. For inverted hierachy, we have :math:`\omega_v < 0`, so that .. math:: \sqrt{ 2\lvert\omega_v\rvert }{ (1+\sqrt{\alpha})^2 } < \mu < \sqrt{ 2\lvert\omega_v\rvert }{ (1-\sqrt{\alpha})^2 }, Within this region, we have exponential growth. Refs & Notes ------------------------- .. [Samuel1996] Samuel, S. (1996). Bimodal coherence in dense self-interacting neutrino gases. Physical Review D, 53(10), 5382–5393. doi:10.1103/PhysRevD.53.5382 .. [Duan2013] Duan, H. (2013). `Flavor oscillation modes in dense neutrino media `_. Physical Review D - Particles, Fields, Gravitation and Cosmology, 88. .. [Wolfenstein1978] Wolfenstein, L. `Neutrino oscillations in matter. `_ *Phys. Rev. D* **17**, 23692374 (1978). Or check papers of MSW effect such as Wick Haxton's excellent review.